Convergence to the planar interface for a nonlocal free‐boundary evolution

Author:

Otto Felix1,Schubert Richard2,Westdickenberg Maria G.3

Affiliation:

1. MPI for Mathematics in the Sciences, Leipzig, Germany

2. University of Bonn, Bonn, Germany

3. RWTH Aachen University, Aachen, Germany

Abstract

AbstractWe capture optimal decay for the Mullins–Sekerka evolution, a nonlocal, parabolic free boundary problem from materials science. Our main result establishes convergence of BV solutions to the planar profile in the physically relevant case of ambient space dimension three. Far from assuming small or well‐prepared initial data, we allow for initial interfaces that do not have graph structure and are not connected, hence explicitly including the regime of Ostwald ripening. In terms only of initially finite (not small) excess mass and excess surface energy, we establish that the surface becomes a Lipschitz graph within a fixed timescale (quantitatively estimated) and remains trapped within this setting. To obtain the graph structure, we leverage regularity results from geometric measure theory. At the same time, we extend a duality method previously employed for one‐dimensional PDE problems to higher dimensional, nonlocal geometric evolutions. Optimal algebraic decay rates of excess energy, dissipation, and graph height are obtained.

Funder

Max-Planck-Institut für Mathematik in den Naturwissenschaften

Deutsche Forschungsgemeinschaft

Publisher

Wiley

Reference29 articles.

1. Ostwald ripening for dilute systems under quasistationary dynamics;Alikakos N. D.;Comm. Math. Phys.,2003

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3. Contraction of convex hypersurfaces in Euclidean space;Andrews B.;Calc. Var. Partial Differential Equations,1994

4. Optimal relaxation of bump‐like solutions of the one‐dimensional Cahn‐Hilliard equation;Biesenbach S.;Comm. Partial Differential Equations,2022

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